A-polynomials, Ptolemy equations and Dehn filling

The A-polynomial encodes hyperbolic geometric information on knots and related manifolds.Historically, it has been difficult to compute, and particularly difficult to determine A-polynomials of infinite families of knots.Here, we compute A-polynomials by starting with a triangulation of a manifold, then using symplectic properties of the Neumann-Zagier matrix encoding the gluings to change the basis of the computation.The result is a simplification of the defining equations.We apply this method to families of manifolds obtained by Dehn filling, and show that the defining equations of their A-polynomials are Ptolemy equations which, up to signs, are equations between cluster variables in the cluster algebra of the cusp torus.57K14, 57K31, 57K32; 57K10 1. Introduction 1265 2. From gluing equations to Ptolemy equations via symplectic reduction 1271 3. Dehn fillings and triangulations 1290 4. Example: Dehn-filling the Whitehead link 1309 Appendix A. Nice triangulations of manifolds with torus boundaries 1312 Appendix B. Calculations for some twist knots 1316